9.1 Introduction
So far our concentration has been on linear forms. We now turn to quadratic forms i.e. highest power of some variable is 2 of degree 2. Like linear forms these have a matrix which is symmetric. Consider the function. \(f:\mathbb {R}^2\longrightarrow \mathbb {R}\) defined by \(F(x_1,x_2)=a_{11}x_1^2+a_{12}x_1x_2+a_{22}x^2_2\).
This is called the quadratic form in \(\mathbb {R}^2\) and in matrix notation we write
\[Q(X)=(x_1,x_2) \begin {pmatrix} a_{11}&\frac {1}{2}a_{12}\\\\ \frac {1}{2}a_{21}&a_{22}\\ \end {pmatrix} \begin {pmatrix} x_1\\x_2\\ \end {pmatrix} \]
where \(X=(x_1,x_2)\) and \(A\) is real symmetric and unique.
Definition 9.1.1. The quadratic form in \(\mathbb {R}^n\) is the function \(Q:\mathbb {R}^n\longrightarrow \mathbb {R}\) defined by
\[Q(X)=\sum ^n_{ij=1}a_{ij}x_ix_j\]
where \(X=(x_1,x_2,\dots ,x_n)\) and \(A=(a_{ij})\) is real symmetric and unique.
In matrix form
\[ Q(X)=(x_1,x_2,\dots ,x_n) \begin {pmatrix} a_{11}&\frac {1}{2}a_{12}&\cdots&a _{1n}\\\\ \frac {1}{2}a_{21}&a_{22}&\cdots& \frac {1}{2}a_{2n}\\\\ \vdots& \vdots& \ddots& \vdots \\ \frac {1}{2}a_{n1}&\frac {1}{2}a_{n2}&\cdots&a _{nn}\\ \end {pmatrix} \begin {pmatrix} x_1\\x_2\\\vdots \\x_n\\ \end {pmatrix} =X^tAX \]
Definition 9.1.2. Express \(Q(x_1,x_2)=2x^2_1+3x^2_2+2x_1x_2\) in matrix form. \[Q(x_1,x_2)=(x_1,x_2) \begin {pmatrix} 2&1\\1&3\\ \end {pmatrix} \begin {pmatrix} x_1\\x_2 \end {pmatrix} \] So that \(A= \begin {pmatrix} 2&1\\1&3\\ \end {pmatrix} \).
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